This Circle Is Centered At The Point \[$(3,2)\$\], And The Length Of Its Radius Is 5. What Is The Equation Of The Circle?A. \[$(x^2-3)+(y^2-2)=5^2\$\]B. \[$(x-2)^2+(y-3)^2=25\$\]C. \[$(x-3)^2+(y-2)^2=25\$\]D.
Introduction
In mathematics, a circle is a set of points that are equidistant from a central point known as the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore the equation of a circle and how to derive it using the given information.
What is the Equation of a Circle?
The equation of a circle is a mathematical expression that describes the set of points that lie on the circle. It is typically written in the form:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Deriving the Equation of a Circle
To derive the equation of a circle, we need to use the distance formula. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) is the center of the circle and (x2, y2) is any point on the circle.
In this case, we are given that the center of the circle is at the point (3, 2) and the length of its radius is 5. We can use the distance formula to derive the equation of the circle.
Step 1: Write the Distance Formula
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) is the center of the circle and (x2, y2) is any point on the circle.
Step 2: Substitute the Given Values
We are given that the center of the circle is at the point (3, 2) and the length of its radius is 5. We can substitute these values into the distance formula:
d = √((x2 - 3)^2 + (y2 - 2)^2)
Step 3: Set the Distance Equal to the Radius
Since the distance from the center to any point on the circle is equal to the radius, we can set the distance equal to 5:
√((x2 - 3)^2 + (y2 - 2)^2) = 5
Step 4: Square Both Sides
To eliminate the square root, we can square both sides of the equation:
(x2 - 3)^2 + (y2 - 2)^2 = 25
Step 5: Simplify the Equation
We can simplify the equation by expanding the squared terms:
(x2 - 3)^2 = (x2 - 3)(x2 - 3) (y2 - 2)^2 = (y2 - 2)(y2 - 2)
Expanding the squared terms, we get:
x2^2 - 6x2 + 9 + y2^2 - 4y2 + 4 = 25
Step 6: Combine Like Terms
We can combine like terms by adding or subtracting the coefficients of the same variables:
x2^2 - 6x2 + y2^2 - 4y2 = 12
Step 7: Write the Equation in Standard Form
We can write the equation in standard form by rearranging the terms:
(x2 - 3)^2 + (y2 - 2)^2 = 25
Conclusion
In this article, we have derived the equation of a circle using the given information. We have used the distance formula to find the equation of the circle and have simplified it to the standard form. The equation of the circle is:
(x - 3)^2 + (y - 2)^2 = 25
This equation describes the set of points that lie on the circle with center (3, 2) and radius 5.
Answer
The correct answer is:
Q: What is the equation of a circle?
A: The equation of a circle is a mathematical expression that describes the set of points that lie on the circle. It is typically written in the form:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Q: How do I derive the equation of a circle?
A: To derive the equation of a circle, you need to use the distance formula. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) is the center of the circle and (x2, y2) is any point on the circle.
Q: What is the distance formula?
A: The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Q: How do I use the distance formula to find the equation of a circle?
A: To use the distance formula to find the equation of a circle, you need to substitute the given values into the formula. For example, if the center of the circle is at the point (3, 2) and the length of its radius is 5, you can substitute these values into the distance formula:
d = √((x2 - 3)^2 + (y2 - 2)^2)
Q: What is the standard form of the equation of a circle?
A: The standard form of the equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Q: How do I simplify the equation of a circle?
A: To simplify the equation of a circle, you need to expand the squared terms and combine like terms. For example, if the equation of the circle is:
(x2 - 3)^2 + (y2 - 2)^2 = 25
You can simplify it by expanding the squared terms and combining like terms:
x2^2 - 6x2 + y2^2 - 4y2 = 12
Q: What is the center of a circle?
A: The center of a circle is the point that is equidistant from all points on the circle. It is typically denoted by the coordinates (h, k).
Q: What is the radius of a circle?
A: The radius of a circle is the distance from the center of the circle to any point on the circle. It is typically denoted by the variable r.
Q: How do I find the equation of a circle with a given center and radius?
A: To find the equation of a circle with a given center and radius, you need to use the standard form of the equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Q: What are some common mistakes to avoid when deriving the equation of a circle?
A: Some common mistakes to avoid when deriving the equation of a circle include:
- Not using the correct formula for the distance between two points
- Not substituting the given values into the formula correctly
- Not expanding the squared terms correctly
- Not combining like terms correctly
Conclusion
In this article, we have answered some frequently asked questions about the equation of a circle. We have covered topics such as the distance formula, the standard form of the equation of a circle, and how to simplify the equation of a circle. We hope that this article has been helpful in answering your questions about the equation of a circle.